How to maintain passwords when traveling?

I'm traveling out of the country and I don't want to bring my laptop, because I would then worry about it getting stolen or broken. I am bringing an old iPad so that I can check email and look up restaurants, etc. while on the trip, but if it got destroyed it wouldn't be a big deal.

I don't plan to access my bank or other secured websites, but I'm thinking that I might need to if there were some kind of emergency that I needed to deal with. But my bank website has a unique password (plus two factor security) and that password is written down in a file on my computer and not synced to any cloud service.

If I don't bring a copy of that file, I don't know the password (it's random gibberish, to prevent standard exploits). But if I do bring a copy of that file, and anyone got that copy away from me, it would be horrific, because they would get passwords to everything.

So how could I work this? I could put the file on a USB stick, with encryption, but since I won't have my own computer, I can't read it without trusting someone else's, and that would be a huge risk.

I could put the file on the iPad itself, but then if that got stolen, I would worry that someone could access the file. (For that matter, how can I secure the email app on the iPad?)

What is a good way to handle this situation?